basic background of firm s analysis short run production function firm s production in long run, firm s equlibrium firm s equilibrium upon different

3. PRODUCTION ANALYSIS OF THE FIRM

3. PRODUCTION ANALYSIS OF THE FIRM

CONTENTS CONTENTS

 basic background of firm´s analysis

 short run production function

 firm´s production in long run, firm´s equlibrium

 firm´s equilibrium upon different levels of total costs, and prices of inputs

 returns to scale

 examples of long run production functions

BASIC BACKGROUND OF FIRM

BASIC BACKGROUND OF FIRM´´S ANALYSIS S ANALYSIS

 firm = subject producing goods and/or services...

transformig inputs into outputs

 firm: recruits the inputs

organizes their transformation into outputs

sells its outputs

 firm´s goal is to maximize its profit

 economic vs. accountable profit

 ekonomic profit = accountable profit minus implicit

costs

BASIC BACKGROUND OF FIRM

BASIC BACKGROUND OF FIRM´´S ANALYSIS S ANALYSIS

 production limits – technological and financial

 production function – relationship between the volume of inputs and volume of outputs

 conventional inputs: labour (L), capital (K)

 other inputs: land (P), technological level (τ)

 production function: Q = f(K,L)

 short run – volume of capital is fixed

 long run – all inputs are variable

SHORT RUN PRODUCTION FUNCTION SHORT RUN PRODUCTION FUNCTION

A

A' B'

B C

C'

L

L TP_L

MP_L AP_L

MP_L

AP_L

to A – increasing returns to labour (MP_L increasing)

to B – 1st stage of production –

average product of labour and capital are increasing; motivation to increase production, fixed input not fully utilized

between B and C – 2nd stage of production – average product of labour decreasing, but AP of capital still increasing

behind C – 3rd stage of production – both APs decreasing, total product decreasing

firm endeavours the 2nd stage of production

TP_L

WHAT INFLUENCES THE SHAPE OF MP WHAT INFLUENCES THE SHAPE OF MP _LL ??

 MP

= product of additional unit of labour

 we add: a) additional working hours... or b) additional workers?

 a): MP

influenced with human´s nature

 b): MP

influenced with the nature of

production

SR PRODUCTION

SR PRODUCTION – – SOME IDENTITIES SOME IDENTITIES

 Q = f (K

, L)

 AP

= Q/L AP

= Q/K

 MP

= ∂Q/∂L MP

= ∂Q/∂K

SR PRODUCTION

SR PRODUCTION – – INCREASING RETURNS INCREASING RETURNS TO LABOUR

TO LABOUR

Q

L L

AP_L MP_L

AP_L MP_L TP

Total product increases with increasing rate – TP grows faster than the volume of labour recruited

SR PRODUCTION

SR PRODUCTION – – CONSTANT RETURNS CONSTANT RETURNS TO LABOUR

TO LABOUR

Q

L L

AP_L MP_L

AP_L = MP_L TP

TP increases with constant rate – as fast as volume of labour recruited

SR PRODUCTION

SR PRODUCTION – – DIMINISHING RETURNS DIMINISHING RETURNS TO LABOUR

TO LABOUR

L L

AP_L MP_L

MP_L TP

AP_L Q

TP increases with decreasing rate – TP grows slower than the volume of labour recruitet

PRODUCTION IN LONG RUN (LR) PRODUCTION IN LONG RUN (LR)

 both inputs, labour and capital are variable

 Q = f(K,L)

 LR production function as a „map“ of isoquants

 isoquant = a curve that represents the set of different combination of inputs leading to the

constant volume of total product (output) – analogy

to consumer´s IC

LR PRODUCTION FUNCTION

LR PRODUCTION FUNCTION – – MAP OF ISOQUANTS MAP OF ISOQUANTS

0 L

K

Q₁

Q₂

Q₃

If both inputs are normal, total product grows with both inputs increasing

ISOQUANTS CHARACTERISTICS ISOQUANTS CHARACTERISTICS

 analogy to ICs

 ... are aligned from the cardinalistic point of view (total product is measureable)

 ... do not cross each other

 ... are generally convex to the origin (a firm

usually needs both inputs)

MARGINAL RATE OF TECHNICAL SUBSTITUTION MARGINAL RATE OF TECHNICAL SUBSTITUTION

 MRTS – ratio expressing the firm´s

possibility to substitute inputs with each other, total product remaining constant (compare with consumer´s MRS

)

 MRTS = -ΔK/ΔL

 -ΔK.MP

= ΔL.MP

→ -ΔK/ΔL=MP

/MP

→

MRTS = MP

/MP

ELASTICITY OF SUBSTITUTION ELASTICITY OF SUBSTITUTION

 relative change of ratio K/L to relative change of MRTS

 implies the shape of isoquants

 σ = d(K/L) / ^K/L

dMRTS / ^MRTS

 σ = ∞ for perfectly substituteable inputs

 σ = 0 for perfect complementary inputs

OPTIMAL COMBINATION OF INPUTS OPTIMAL COMBINATION OF INPUTS

 again analogy of consumer´s equilibrium

 firm is limited with its budget

 budget constraint depends on total firm´s

expenditures (total costs – TC), and prices of inputs

 firm´s budget line (isocost):

TC = w.L + r.K, where w…wage rate

r…rental (derived from interest rate)

OPTIMAL COMBINATION OF INPUTS

OPTIMAL COMBINATION OF INPUTS – – INNER SOLUTION INNER SOLUTION

 if isoquant tangents the isocost:

 if the slope of isoquant equals the slope of isocost...

 ...if stands: MRTS = w/r , so:

 MP

/MP

= w/r

 only in the above case the firm produces the specific output with minimal total costs, or:

 ... firm produces maximal output with specific

level of total costs

OPTIMAL COMBINATION OF INPUTS

OPTIMAL COMBINATION OF INPUTS – – INNER INNER SOLUTION

SOLUTION

E

L K

L*

K* B

A firm´s

equilibrium

In A and B, the firm is not minimizing its total costs

In A and B, the firm is not maximizing its total product Q

TC₁ TC₂

OPTIMAL COMBINATION OF INPUTS

OPTIMAL COMBINATION OF INPUTS – – CORNER SOLUTION CORNER SOLUTION

 usually in the case of perfectly substituteable inputs... then...

 ...firm´s equilibrium as an intersection of isoquant and isocost

 MRTS ≠ w/r

OPTIMAL COMBINATION OF INPUTS

OPTIMAL COMBINATION OF INPUTS – – CORNER SOLUTION CORNER SOLUTION

E

L K

K* firm´s

equilibrium

TC Q

E L K

L*

firm´s equilibrium Q

TC

MRTS < w/r MRTS > w/r

COST EXPANSION PATH (CEP) COST EXPANSION PATH (CEP)

 set of firm´s equilibria upon different levels of total costs (budgets)

 analogy to consumer´s ICC

L K

E₁

E₂

E₃

CEP

PRICE EXPANSION PATH PRICE EXPANSION PATH



set of firm´s equilibria upon different levels of price of one of the inputs



analogy to consumer´s PCC

E₁

E₂

E₃ PEP

L K

IMPACT OF THE CHANGE OF PRICE OF INPUT ON THE IMPACT OF THE CHANGE OF PRICE OF INPUT ON THE QUANTITY IN PRODUCTION PROCESS

QUANTITY IN PRODUCTION PROCESS – – SUBSTITUTION AND PRODUCT

SUBSTITUTION AND PRODUCTION ION EFFECT EFFECT

 substitution effect (SE) – the firm

substitutes relatively more costly input with the relatively cheaper one

 production effect (PE) – analogy to

consumer´s IE – change of price of input

leads to the change of real budget

L K

Q₁

Q₂

SE PE

A

TE B

C

Shift from A to B – substitution effect – total product remains constant (we use

Hicks´s approach

Shift from B to C – production effect – total product increases

Shift from A to C – total effect – the sum of SE and PE

SUBSTITUTION AND PRODUCTION EFFECT OF WAGE SUBSTITUTION AND PRODUCTION EFFECT OF WAGE RATE DECREASE

RATE DECREASE

TC₂ TC₁

RETURNS TO SCALE RETURNS TO SCALE

 we compare the relative change of total product and relative change of inputs

volume

 diminishing, constant or increasing

 diminishing: total product grows slower than the volume of inputs recruited

 constant: total product and the volume of inputs grow by the same rate

 increasing: total product grows faster

than the volume of inputs recruited

LR

LR PRODUCTION FUNCTION PRODUCTION FUNCTIONS S UPON DIFFERENT TYPES OF UPON DIFFERENT TYPES OF RETURNS TO SCALE

RETURNS TO SCALE

Q=10 Q=10

Q=20 Q=30

Q=20 Q=90

Q=10

K K K Q=20

L L L

constant – isoquants keep

the same distance increasing – isoquants

get closer diminsing – isoquants draw apart from each

other

EXAMPLES OF PRODUCTION FUNCTIONS EXAMPLES OF PRODUCTION FUNCTIONS

1. Linear production function Q = f(K,L) = a.K + b.L

 contents constant returns to scale:

f(t.K,t.L) = a.t.K + b.t.L = t(a.K+b.L) = t.f(K,L)

 elasticity of inputs substitution:

σ = ∞ → labour and capital are perfect

substitutes

EXAMPLES OF PRODUCTION FUNCTIONS EXAMPLES OF PRODUCTION FUNCTIONS

2. With fixed inputs proportion:

Q = min(a.K,b.L)

„min“ says that total product is limited with smaller value of one of the inputs – i.e. 1 lory needs 1 driver – if we add second driver, we do not raise the total volume of transported load

 contents constant returns to scale:

f(t.K,t.L) = min(a.t.K,b.t.L) = t.min(a.K,b.L) = t.f(K,L)

 elasticity of inputs substitution:

σ = 0 → labour and capital are perfect

complements

EXAMPLES OF PRODUCTION FUNCTION EXAMPLES OF PRODUCTION FUNCTION

3. Cobb-Douglas production function:

Q = f(K,L) = A.K

.L

 returns to scale:

f(t.K,t.L) = A.(t.K)

(t.L)

= A.t

.K

.L

= t

.f(K,L) depend on the value of “a“ and “b“, if:

a+b=1 → constant

a+b>1 → increasing

a+b<1 → diminishing

EXAMPLES OF PRODUCTION FUNCTIONS EXAMPLES OF PRODUCTION FUNCTIONS

Q₃ Q₂ Q₁

K K

L L

K

L

Q₁ Q₁

Q₂ Q₂

Q₃

Q₃

Linear production

funcstion Production function with fixed proportion

of inputs

Cobb-Douglas production function